Consider the following statements in respect of any triangle
I. The three medians of a triangle divide it into six triangles of equal area.
II. The perimeter of a triangle is greater than the sum of the lengths of its three medians.
Which of the statements given above is/are correct ?

Options

A. I only
B. II only
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to evaluate the two statements regarding the properties of a triangle and its medians.

### Step 1: Analyze Statement I
**Statement I:** The three medians of a triangle divide it into six triangles of equal area.

- A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
- When all three medians are drawn, they intersect at a point called the centroid.
- The centroid divides each median into two segments, with the segment connecting the centroid to the vertex being twice as long as the segment connecting the centroid to the midpoint of the opposite side.
- While the medians do divide the triangle into six smaller triangles, these triangles are not necessarily of equal area unless the triangle is equilateral. For scalene or isosceles triangles, the areas of the resulting triangles will differ.

**Conclusion for Statement I:** This statement is **false**.

### Step 2: Analyze Statement II
**Statement II:** The perimeter of a triangle is greater than the sum of the lengths of its three medians.

- The perimeter of a triangle is the sum of the lengths of its three sides (let's denote them as \( a, b, c \)).
- The lengths of the medians can be calculated using the formula for the length of a median \( m_a \) from vertex A to side BC:
\[ m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} \]
- By applying the triangle inequality, we can show that the sum of the lengths of the sides of the triangle is greater than the sum of the lengths of the medians. Specifically, the perimeter \( P = a + b + c \) is indeed greater than \( m_a + m_b + m_c \).

**Conclusion for Statement II:** This statement is **true**.

### Final Conclusion
- Statement I is false.
- Statement II is true.
- Therefore, the correct answer is that only Statement II is correct.

### Summary of the Solution
- **Statement I:** False
- **Statement II:** True
- **Correct Answer:** Only Statement II is correct.

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